Inspired by the relation between deep neural network (DNN) and partial
differential equations (PDEs), we study the general form of the PDE models of
deep neural networks. To achieve this goal, we formulate DNN as an evolution
operator from a simple base model. Based on several reasonable assumptions, we
prove that the evolution operator is actually determined by
convection-diffusion equation. This convection-diffusion equation model gives
mathematical explanation for several effective networks. Moreover, we show that
the convection-diffusion model improves the robustness and reduces the
Rademacher complexity. Based on the convection-diffusion equation, we design a
new training method for ResNets. Experiments validate the performance of the
proposed method.

深度神经网络与偏微分方程之间的关系启发我们研究了深度神经网络的偏微分方程模型的一般形式。在一些合理的假设下，我们证明演化算子实际上由对流 - 扩散方程决定。这个模型为几个有效的网络提供了数学解释。此外，我们还展示了对流 - 扩散模型提高了鲁棒性并减小了 Rademacher 复杂度。基于对流 - 扩散方程，我们设计了一种适用于 ResNets 的新训练方法。实验证实了所提出方法的性能。